REACTIVE SOLUTE TRANSFER IN MAGNETOHYDRODYNAMIC BOUNDARY LAYER STAGNATION-POINT FLOW OVER A...

Reactive Solute Transfer in MagnetohydrodynamicBoundary Layer Stagnation-Point Flow over a

Stretching Sheet with Suction/Blowing

KRISHNENDU BHATTACHARYYA,SWATI MUKHOPADHYAY, AND G. C. LAYEK

Department of Mathematics, The University of Burdwan, Burdwan,West Bengal, India

The article is concerned with the study of reactive solute distribution in the laminarboundary layer stagnation-point flow of an electrically conducting viscous incom-pressible fluid over a stretching sheet with suction or blowing. The fluid is permeatedby a magnetic field produced by an electric current. The self-similar equations formomentum and mass transfer are obtained and solved numerically by finite differ-ence method using quasi-linearization technique. The velocity and reactive concen-tration profiles for various values of parameters involved in the system arepresented through graphs and analyzed physically. They reveal that the momentumand solute boundary layer thicknesses decrease with increasing values of b=a< 1(velocity ratio parameter) as well as b=a> 1. Furthermore, it is noted that withthe increase of Hartmann number, the solute transfer increases when b=a< 1 anddecreases when b=a> 1, but the opposite behavior has been noticed in the case ofvelocity profiles.

Keywords Magnetohydrodynamic boundary layer; Reactive solute; Stagnationpoint; Stretching sheet; Suction=blowing

Introduction

The laminar boundary layer stagnation-point flow over a stretching sheet hasreceived considerable attention due to its enormous engineering applications. Thediffusion of contaminants in boundary layer flow around a stretching sheet is rel-evant practically in chemical industries and metal and polymer processing industriesand also for environmental pollution, biological processes, and many other areas.Laminar boundary layer flow driven by a continuously moving solid surface wasfirst studied by Sakiadis (1961a, 1961b). Crane (1970) gave an exact similarity sol-ution in closed analytical form for steady boundary layer flow of an incompressibleviscous fluid past a stretching plate. Heat and mass transfer analysis for Newtonianboundary layer flow on a stretching sheet with suction or blowing was investigatedby Gupta and Gupta (1977). Chakrabarti and Gupta (1979) analyzed

Address correspondence to G. C. Layek, Department of Mathematics, The University ofBurdwan, Burdwan-713104, West Bengal, India. E-mail: [email protected]

Chem. Eng. Comm., 199:368–383, 2012Copyright # Taylor & Francis Group, LLCISSN: 0098-6445 print=1563-5201 onlineDOI: 10.1080/00986445.2011.592444

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magnetohydrodynamic (MHD) flow over a stretching sheet at a different value ofthe parameter related to uniform temperature.

The boundary layer stagnation-point flow over a stretching sheet in various con-texts has been investigated by many authors. Chiam (1994) studied thetwo-dimensional steady stagnation-point flow of an incompressible viscous fluidtowards a stretching surface, but found no boundary layer. Mahapatra and Gupta(2001, 2002) reconsidered the stagnation-point flow problem towards a stretchingsurface with a uniform transverse magnetic field and heat transfer with viscous dis-sipation effects and observed boundary layer structure near the surface. Pop et al.(2004) discussed the flow over a stretching sheet near a stagnation point consideringradiation effects, and Layek et al. (2007) investigated boundary layer stagnation-point flow towards a porous stretching sheet with heat absorption=generation.Recently, the effects of thermal conductivity and heat source=sink on MHD flownear a stagnation point on a linearly stretching sheet was studied by Sharma andSingh (2009).

The effects of chemical reaction for concentration distribution on fluid flow dueto a stretching sheet were studied by many researchers with several physical aspects.Chambre and Young (1958) analyzed the diffusion of a chemically reactive species inlaminar boundary layer flow. Andersson et al. (1994) discussed the transfer of che-mically reactive species in laminar flow over a stretching sheet. The similarity solu-tions of mixed convective flow with heat and mass transfer over a horizontal movingplate with chemical reaction were obtained by Fan et al. (1998). Anjalidavi andKandasamy (1999, 2000) studied the effects of chemical reaction and heat and masstransfer on laminar flow along a semi-infinite horizontal plate, and later they alsoanalyzed the effects of magnetic field. The effects of chemical reaction on MHD freeconvective flow and mass transfer over a stretching sheet was investigated by Afify(2004). Akyildiz et al. (2006) discussed the diffusion of chemically reactive species fornon-Newtonian fluid flow over a stretching sheet immersed in a porous medium.Cortell (2007) studied motion and mass transfer with chemically reactive speciesfor two classes of viscoelastic fluid over a porous stretching sheet. Kechil andHashim (2008) obtained the series solution of the boundary layer flow over a non-linearly stretching sheet in the presence of chemical reaction and magnetic fieldemploying the Adomian decomposition method. Recently, Chamkha et al. (2010)analyzed the effects of chemical reaction on unsteady free convective heat and masstransfer on a porous stretching surface in a porous medium, and El-Aziz (2010) stud-ied unsteady flow due to a stretching sheet with mass and heat transfer and chemicalreaction.

Mass transfer over a stretching sheet near a stagnation point not yet been stud-ied. For this, in the present investigation, we have investigated chemically reactivesolute transfer in a steady Newtonian MHD boundary layer stagnation-point flowover a stretching sheet with suction or blowing. First-order chemical reaction andthe variable solute distribution along the sheet are taken into account. Using thesimilarity solution technique, the partial differential equations governing the motionand the concentration distribution are transformed into self-similar ordinary differ-ential equations and are then solved numerically by the finite difference methodusing the quasi-linearization technique. The results are presented through graphsand discussed physically for several variations of parameters involved in theequations.

Reactive Solute Transfer in MHD 369

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Mathematical Formulation of the Flow Problem

Let us consider the laminar stagnation-point flow of an incompressible Newtonianelectrically conducting fluid along a stretching sheet with chemically reactive solutedistributed in the fluid. Using boundary layer approximations, the equation ofmotion for steady two-dimensional flow in presence of an externally applied mag-netic field directed normal to the plane of flow and the equation representing reactivesolute are written in the usual notation as follows:

@u

@xþ @v

@y¼ 0 ð1Þ

u@u

@xþ v

@u

@y¼ � 1

q@p

@xþ t

@2 u

@y2þ 1

qðJ� BÞx ð2Þ

u@C

@xþ v

@C

@y¼ D

@2C

@y2� kðC � C1Þ ð3Þ

where u and v are velocity components in the x- and y-directions respectively, t is thekinematic fluid viscosity, q is the fluid density, p is the pressure, J is the current den-sity, B is the externally applied magnetic field, C is the concentration, D is the solutediffusion coefficient, and k is the reaction rate of the solute. In writing Equation (2),we have neglected the induced magnetic field, assuming the magnetic Reynoldsnumber for the flow to be small.

The applied magnetic field B has components (0, B0, 0) with B0 (�0) being con-stant and, consequently, the relation $ �B¼ 0 is automatically satisfied. Here, theelectric current in the flow acts parallel to the z-axis. Hence from Ohm’s law weget the components of J as

jx ¼ 0; jy ¼ 0; jz ¼ r½Ez þ ðq� BÞz� ¼ r½Ez þ uB0� ð4Þ

where r is the electrical conductivity of the conducting fluid, Ez is the component ofthe electric field E along the z-direction, and q is the velocity vector. Now, as the flowis steady, Maxwell’s equation gives

r� E ¼ 0 ð5Þ

where E is along the z-axis. This gives from Equation (5) @Ez=@y¼ 0 and @Ez=@x¼ 0,so that Ez is a function of z only.

Since the induced magnetic field is neglected, electric current in the flow is deter-mined from Ohm’s law and not from $�B¼ leJ, le being the magnetic per-meability. But the consequence $ � J¼ 0 of this equation must be satisfied. Thisreadily gives from Equation (4) that Ez is constant, since Ez is independent of xand y. Thus using Equation (4), we find from Equation (2)

u@u

@xþ v

@u

@y¼ � 1

q@p

@xþ t

@2 u

@y2� rB2

0

qu� rB0Ez

qð6Þ

370 K. Bhattacharyya et al.

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In the inviscid free stream, u¼U1(x) and we obtained from Equation (6) as

U1dU1dx

¼ � 1

q@p

@x� rB2

0

qU1 � rB0Ez

qi:e: � 1

q@p

@x� rB0Ez

q¼ U1

dU1dx

þ rB20

qU1

Here, U1 is the free-stream velocity and in the present problem U1¼ bx.Then the momentum equation (6) reduces to

u@u

@xþ v

@u

@y¼ b2xþ t

@2 u

@y2� rB2

0

qðu� bxÞ ð7Þ

The boundary conditions for the velocity field and concentration distribution aregiven by

u ¼ Uw ¼ ax; v ¼ �vw at y ¼ 0; u ! U1 ¼ bx as y ! 1 ð8Þ

C ¼ Cw ¼ C1 þ C0xn at y ¼ 0; C ! C1 as y ! 1 ð9Þ

where a and b are the stretching and the straining parameters, respectively, and areassumed to be constant with a> 0 and b> 0. We also consider a variable solute dis-tribution along the stretching sheet, i.e., Cw¼C1þC0x

n, where C1 is constant sol-ute retaining at large y and C0> 0 is constant; n is a power-law exponent signifyingthe change of amount of solute in the x-direction.

Here, vw is a distribution of suction=blowing, which is prescribed as follows:

vw ¼ sðatÞ1=2

where s is the suction=blowing parameter measuring the strength of the appliedsuction=blowing through the stretching sheet, i.e., when s> 0 it is suction and whens< 0 it is blowing.

A brief inspection provides us a set of similarity solutions of the above system ofequations as

u ¼ axf 0ðgÞ; v ¼ �ðatÞ1=2f ðgÞ and C ¼ C1 þ /ðgÞðCw � C1Þ

where g is the similarity variable and is given by g¼ y(a=t)1=2.For the above expression of u and v the continuity equation (1) satisfied identi-

cally. Also, substituting the values of u, v, and C in Equations (7) and (3) we obtainthe reduced self-similar equations as

f 000 þ ff 00 � f 02 �M2 f 0 þM2b=aþ b2=a2 ¼ 0 ð10Þ

/00 þ Sc f/0 � Scðn f 0 þ bÞ/ ¼ 0 ð11Þ

where M ¼ ðrB20=aqÞ

1=2 is the Hartmann number, Sc¼ t=D is the Schmidt number,and b¼ k=a is the reaction- rate parameter.

Also, the boundary conditions (8) and (9) are transformed to the followingforms:

f ðgÞ ¼ s; f 0ðgÞ ¼ 1 at g ¼ 0; f 0ðgÞ ! b=a as g ! 1 ð12Þ


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/ðgÞ ¼ 1 at g ¼ 0; /ðgÞ ! 0 as g ! 1 ð13Þ

where b=a is velocity ratio parameter.The shear stress at the sheet is denoted by sw and is defined as

sw ¼ l@u

@y

� �y¼0

¼ l½ax f 00ðgÞffiffiffiffiffiffiffia=t

p�y¼0 ¼ l

ffiffiffiffiffiffiffia=t

pðaxÞf 00ð0Þ

We now define the skin friction coefficient (Sw) at the sheet as

Sw ¼ sw1

2qU2

w

� ��

so that we get

1

2SwRe

1=2x ¼ f 00ð0Þ

The rate of mass flux (Rmf) at the wall is defined as

Rmf ¼ �D@C

@y

� �y¼0

¼ �Dffiffiffiffiffiffiffia=t

pC0x

n/0ð0Þ

and the local Sherwood number (Sh) relating to the rate of solute transfer is

Sh ¼ xRmf =fDðCw � C1Þg

Hence we get

ShRe�1=2x ¼ �/0ð0Þ

where Rex¼ xUw=t is the local Reynolds number based on the stretching velocity.These parameters play a very important role in the analysis of the flow behavior

as well as the solute transfer over the stretching sheet near a stagnation point.

Method of Solution

The self-similar nonlinear equations of the dynamical system represented by Equa-tions (10)–(13) are solved numerically by the finite difference method using thequasi-linearization technique (Bellman and Kalaba, 1965).

Applying the quasi-linearization technique, the discretized version of Equations(10) and (11) are written as follows:

F 00ðiþ1Þ þ f ðiÞF 0ðiþ1Þ � ð2F 0ðiÞ þM2ÞF ðiþ1Þ ¼ �F 2ðiÞ �M2b=a� b2=a2 ð14Þ

/00ðiþ1Þ þ Sc f ðiþ1Þ/0ðiþ1Þ � Scðn F ðiþ1Þ þ bÞ/ðiþ1Þ ¼ 0 ð15Þ

where F¼ f0.


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The boundary conditions reduce to the forms given below:

f ðiþ1Þ ¼ s; F ðiþ1Þ ¼ 1 at g ¼ 0; F ðiþ1Þ ¼ b=a at g ¼ g� ð16Þ

/ðiþ1Þ ¼ 1 at g ¼ 0; /ðiþ1Þ ¼ 0 at g ¼ g� ð17Þ

The functions with iteration index (i) denote the i-th iteration level and those withthe index (iþ 1) denote the (iþ 1)-th level; g� is a suitable distance from the originselected by consideration of flow behavior in the outer flow region of the boundarylayer stagnation-point flow over a linearly stretching sheet.

So, we divide the interval [0, g�] into N equal subintervals of length Dg¼ 0.001,taking g� ¼ 5 for all cases under investigation. Applying central finite differenceformulas of the second- and first-order derivatives of F as

F 00 ¼ Fjþ1 � 2Fj þ Fj�1

ðDgÞ2and F 0 ¼ Fjþ1 � Fj�1

2ðDgÞ

and similarly for /, the above system of Equations (14) and (15) along with theboundary conditions (16) and (17) reduce to

Fðiþ1Þj�1 aj þ F

ðiþ1Þj bj þ F

ðiþ1Þjþ1 cj ¼ dj ; 1 � j � N ð18Þ

and

/ðiþ1Þj�1 pj þ /ðiþ1Þ

j qj þ /ðiþ1Þjþ1 rj ¼ 0; 1 � j � N ð19Þ

with

fðiþ1Þ0 ¼ s; F

ðiþ1Þ0 ¼ 1 and F

ðiþ1ÞNþ1 ¼ b=a ð20Þ

and

/ðiþ1Þ0 ¼ 1 and /ðiþ1Þ

Nþ1 ¼ 0 ð21Þ

where

aj ¼1

ðDgÞ2�

fðiÞj

2ðDgÞ ; bj ¼ � 2

ðDgÞ2� 2f

0ðiÞj �M2; cj ¼

1

ðDgÞ2þ

fðiÞj

2ðDgÞ ;

dj ¼ �f2ðiÞj �M2b=a� b2=a2; 1 � j � N;

and

pj ¼1

ðDgÞ2�Sc f

ðiþ1Þj

2ðDgÞ ;


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qj ¼ � 2

ðDgÞ2� Scðn F

ðiþ1Þj þ bÞ; rj ¼

1

ðDgÞ2þSc f

ðiþ1Þj

2ðDgÞ ; 1 � j � N:

We solve the system of algebraic (tri-diagonal system) Equations (18) with theconditions (20) by the standard Thomas algorithm. Using the obtained results for

fðiþ1Þj and F

ðiþ1Þj the system of Equations (19) representing the discretized form of

solute transfer equation with the conditions (21) are solved numerically.

Results and Discussions

Numerical computations have been carried out for various values of the parametersinvolved in these equations, namely, velocity ratio parameter b=a, Hartmann num-ber M, Schmidt number Sc, reaction rate parameter b, power-law exponent n, andthe strength relating suction=blowing parameter s. For illustration of the results,computed values are plotted through graphs, and the physical explanations arerendered for every case.

For the verification of accuracy of the applied numerical scheme, a comparisonof the present results corresponding to the skin-friction coefficient f00(0) for variablevalues of b=a and M in the absence of suction=blowing with the available results ofMahapatra and Gupta (2001) is made and presented in Table I. The results are foundin excellent agreement.

First, we focus our attention on the variation of velocity and solute profiles fordifferent values of the velocity ratio parameter b=a. In Figure 1, the velocity curvesare presented for various values of b=a with other parameters remaining constant.Here three sets of values of b=a, b=a> 1, b=a< 1, and b=a¼ 1, are taken into con-sideration. From the figure, it is noted that when b=a> 1, the boundary layer thick-ness decreases with increasing b=a and similarly when b=a< 1 it also decreases withincreasing b=a, but for b=a¼ 1, there is no boundary layer structure. This result iscompatible with the physical behavior of this flow. Actually, the inverted boundarylayer is developed when the stagnation flow velocity is less than the stretching velo-city. The concentration profiles for several values of b=a are depicted in Figure 2,and it is found that the value of solute distribution, for fixed g, decreases signifi-cantly with the increase in the parameter b=a. It is more important to note that whenb¼ a, there exists a concentration boundary layer, but no velocity boundary layer isformed. In the boundary layer region, the velocity field has a major impact on the

Table I. Values of skin friction coefficient f00(0) for different values of b=a and Mwith s¼ 0

Mahapatra and Gupta (2001) Present study

M b=a¼ 0.1 b=a¼ 2 b=a¼ 0.1 b=a¼ 2

0 �0.9643 1.9990 �0.9648 1.99922 �2.0198 2.7970 �2.0197 2.79713 �2.8222 3.5460 �2.8225 3.54614 �3.6509 4.3773 �3.6509 4.37745 �4.4863 5.2454 �4.4865 5.2456


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diffusion of solute, but the concentration boundary layer is quite different fromthe velocity boundary layer. So, it is obvious that in the absence of velocityboundary layer (i.e., when b¼ a), the formation of concentration boundary layeris possible.

Next, we consider the effects of the applied magnetic field on velocity and con-centration profiles. The velocity curves for various values of Hartmann number Mfor two values of b=a (one is greater than 1, other is less) are plotted in Figure 3.From the figure it is clear that, for b=a< 1 the velocity at a point decreases withthe increase of M and for b=a> 1, velocity increases. So, it is evident that magneticforce opposes the fluid motion when b=a< 1 and assists the motion when b=a> 1.This happens due to the appearance of the Lorentz force in the interaction of theelectrically conducting fluid motion and applied magnetic force. Figure 4 displaysreactive concentration profiles for different values of M. Opposite to the nature ofvelocity distribution, it is seen that for b=a< 1 the value of solute at particular g

Figure 2. Reactive concentration profiles /(g) for various values of b=a with M¼ 1, s¼ 0.5,Sc¼ 1, b¼ 1, n¼ 1.

Figure 1. Velocity profiles f0(g) for various values of b=a with M¼ 1, s¼ 0.5.


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increases with the increase of M. This implies that the magnetic force enhancesdiffusion of the reactive solute from the sheet when b=a< 1. For b=a> 1, the valueof the solute at a fixed g decreases with the increase of M. Hence, in this case mag-netic force opposes the diffusion process of reactant solute. This result may be veryuseful in the situation where the enhancement of solute from the plate is of primeimportance.

We now pay attention to the variation of velocity and the corresponding solutetransfer for various values of suction=blowing parameter s. The velocity profiles forvarious values of s are presented in Figure 5. The figure reveals that for a fixed valueof g and for b=a¼ 0.1, velocity curves decrease when s increases, whereas for b=a¼ 2,velocity curves increase with increasing s. Furthermore, it is observed that blowing(s< 0) acts to increase the velocity boundary layer thickness, whereas suction(s> 0) acts contrarily in both situations, b=a< 1 and b=a> 1. This behavior of thevelocity distribution is reflected in the distribution of reactive solute, as seen in

Figure 4. Reactive concentration profiles /(g) for various values of M and b=a with s¼ 0.5,Sc¼ 1, b¼ 1, n¼ 1.

Figure 3. Velocity profiles f0(g) for various values of M and b=a with s¼ 0.5.


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Figure 6. When parameter s increases, the value of the concentration profilesdecreases at a particular g. Similar to the velocity profile, suction (s> 0) decreasesthe solute boundary layer thickness and blowing (s< 0) increases it.

We now concentrate our attention on the solute diffusion for the variation ofSchmidt number (Sc) with b=a¼ 0.1 and 2. Actually, the Schmidt number is inverselyproportional to the diffusion coefficient (D). So, increasing values of Sc result indecreasing the solute diffusion coefficient. In Figure 7, the curves display the solutedistribution for increasing values of Sc, and they clearly show that for both b=a< 1and b=a> 1, the thickness of concentration boundary layer is decreased due to theincrease of solute diffusion coefficient. At large values of g, the solute profiles, asexpected, vanish.

Next, we discuss the effect of reaction rate parameter on the diffusion of a che-mically reactive solute. Figure 8 is the graphical representation of concentration pro-files for various values of reaction rate parameter b for b=a¼ 0.1 and 2 when

Figure 6. Reactive concentration profiles /(g) for various values of s and b=a with M¼ 1,Sc¼ 1, b¼ 1, n¼ 1.

Figure 5. Velocity profiles f0(g) for various values of s and b=a with M¼ 1.


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the values of other parameters remain fixed. Various curves representing /(g) fordifferent values of b show that the value of reactive solute distribution decreases withincreasing b for both values of b=a. So, in case of the distribution of a reactive solute,the reaction constant parameter is a decelerating agent, and it thins the soluteboundary layer formed in the neighborhood of the wall.

Finally, we consider the effects on solute boundary layer when the initial distri-bution of solute is varied over the stretching sheet. Keeping other parameters con-stant, the concentration profiles for different values of power-law exponent n, fortwo values of b=a (one for b=a> 1 and other for b=a< 1) are depicted in Figure 9.For both values of b=a (0.1 and 2), increase in the value of n decreases the valueof concentration at a particular point, i.e., the diffusivity of the solute decreases withthe increase of wall concentration. So, variable concentration distribution over the

Figure 8. Reactive concentration profiles /(g) for various values of b and b=a with M¼ 1,s¼ 0.5, Sc¼ 1, n¼ 1.

Figure 7. Reactive concentration profiles /(g) for various values of Sc and b=a with M¼ 1,s¼ 0.5, b¼ 1, n¼ 1.


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stretching sheet causes a decrease in the rate of solute transfer in boundary layerflow. It is noteworthy that the wall concentration is constant when n¼ 0.

The momentum and solute boundary layer thicknesses are denoted by d and dC,respectively, and defined as the values of g (distances from the surface) at which thedifference of dimensionless velocity f0 and the parameter b=a and the dimensionlessconcentration / have decayed to 0.001, respectively. The boundary layer parametersd and dC are described by the equations d¼ gd(t=a)

1=2 and dC¼ gdC(t=a)1=2. The

dimensionless thicknesses gd and gdC are given in Table II for several values of b=a. From the table, it is seen that the velocity boundary layer thickness decreases whenb=a increases (for both b=a< 1 and b=a> 1). Similar behavior is found in case ofsolute boundary layer thickness (dC).

Tables III–VII exhibit the effects of parameter b=a, Hartmann number M,suction=blowing parameter s, Schmidt number Sc, reaction-rate parameter b, andpower-law exponent n on the skin friction coefficient f00(0) and local Sherwoodnumber �/0(0). From the results of Table III, it is found that the skin friction coef-ficient increases with increasing M for fixed values of b=a greater or less than 1. Onthe other hand, for a fixed value of M, f00(0) decreases with increase in b=a providedthat b=a< 1, and for b=a> 1, f00(0) increases when b=a increases. As M increases,the Lorentz force, which opposes the flow, also increases, and the boundary layerthickness decreases when b=a (for b=a> 1) increases. Table IV shows that localSherwood number decreases with increasing M for a fixed value of b=a< 1, but

Figure 9. Reactive concentration profiles /(g) for various values of n and b=a with M¼ 1,s¼ 0.5, Sc¼ 1, b¼ 1.

Table II. Values of gd and gdC for several values of b=a with M¼ 1, s¼ 0.5, Sc¼ 1,b¼ 1, n¼ 1

b=a 0.1 0.3 0.5 1.5 2.0 2.5

gd 3.64 3.01 2.63 1.92 1.87 1.79gdC 3.775 3.292 3.001 2.198 1.986 1.827


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Table III. Values of skin friction coefficient f00(0) for several values of b=a and Mwith s¼ 0.5

M b=a! 0.1 0.2 0.5 2.0 3.0

0.0 �1.224399 �1.148939 �0.818342 2.302825 5.3095220.5 �1.318604 �1.227409 �0.859916 2.361834 5.4099561.0 �1.567036 �1.437008 �0.974357 2.531886 5.7032101.5 �1.901984 �1.724630 �1.139820 2.787471 6.1562652.0 �2.281457 �2.054960 �1.334965 3.110170 6.737768

Table IV. Values of local Sherwood number �/0(0) for several values of b=a and Mwith s¼ 0.5, Sc¼ 1, b¼ 1, n¼ 1

M b=a! 0.1 0.2 0.5 2.0 3.0

0.0 1.736015 1.748234 1.794696 2.069890 2.2412540.5 1.726717 1.741409 1.792014 2.071708 2.2436381.0 1.705457 1.724571 1.784801 2.078056 2.2526981.5 1.679093 1.703799 1.775682 2.086818 2.2647982.0 1.653105 1.682430 1.765191 2.096713 2.279252

Table V. Values of skin friction coefficient f00(0) for several values of b=a and s withM¼ 1

s b=a! 0.1 0.2 0.5 2.0 3.0

�1.0 �0.939757 �0.870943 �0.606984 1.756430 4.117489�0.5 �1.112133 �1.027435 �0.710159 1.985967 4.5982600.0 �1.320541 �1.215607 �0.833392 2.245605 5.1270120.5 �1.567036 �1.437008 �0.974357 2.531886 5.7032101.0 �1.849443 �1.688540 �1.134008 2.839446 6.320595

Table VI. Values of local Sherwood number �/0(0) for several values of b=a and swith M¼ 1, Sc¼ 1, b¼ 1, n¼ 1

s b=a! 0.1 0.2 0.5 2.0 3.0

�1.0 1.008838 1.018733 1.053840 1.277894 1.428783�0.5 1.197428 1.211613 1.257002 1.511752 1.6739960.0 1.429736 1.446485 1.500725 1.778841 1.9489820.5 1.705468 1.724571 1.784801 2.078056 2.2526981.0 2.020717 2.042443 2.105504 2.404839 2.580762


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it increases when M increases for fixed b=a> 1. With an increase in M, the thick-ness of the concentration boundary layer increases for b=a< 1. It is seen fromTables V and VI that f00(0) and �/0(0) increase with increasing suction for all fixedvalues of b=a, and for increasing blowing, their values decrease. This feature alsosupported the earlier findings in Figure 5. Due to suction, concentration layerdecreases. This is consistent with the findings of Figure 6. Finally, Table VII showsthat the local Sherwood number increases when any one of the parameters Sc, b,and n increases for a particular value of b=a. Actually, the Schmidt number isinversely proportional to the diffusion coefficient. The reaction-rate parameter bis a decelerating agent, and as a result, the solute boundary layer near the wallbecomes thinner. This is due to the fact that the conversion of the species takesplace as a result of chemical reaction. For large values of n, a smaller diffusionlayer is expected. This increase in reaction order at fixed Schmidt number isassociated with reduced convection layer.

Conclusions

Chemically reactive solute distribution with first-order reaction in a steady MHDboundary layer stagnation-point flow past a permeable stretching sheet with suctionor blowing is investigated in this work. The power-law distribution of initial soluteover the stretching sheet is considered. This problem (i.e., mass transfer) has notbeen solved before, since previous researchers did not study flow near thestagnation-point region. The self-similar equations obtained for this flow andreactive concentration are solved numerically by the finite difference method usingthe quasi-linearization technique. Our study reveals that the thickness of velocityboundary layer decreases with increase in magnetic parameter and applied suction,whereas it increases with applied blowing. The solute boundary layer thicknessdecreases with increasing values of Schmidt number, reaction-rate parameter,power-law exponent of solute distribution at the sheet, and applied suction, but itincreases with increase of blowing. The skin friction coefficient increases withincreasing values of the magnetic and suction parameters and decreases with increas-ing blowing parameter. The local Sherwood number increases with increasing valuesof Schmidt number, reaction-rate parameter, power-law exponent of initial variablesolute distribution, and suction but decreases with increasing values of the magneticand blowing parameters.

Table VII. Values of local Sherwood number �/0(0) for several values of b=a, Sc, b,and n with M¼ 1, s¼ 0.5

Sc b n b=a! 0.1 0.5 2.0

0.5 1.0 1.0 1.105577 1.180619 1.4330151.0 1.0 1.0 1.705466 1.784801 2.0780562.0 1.0 1.0 2.662450 2.743721 3.0696391.0 0.0 1.0 1.253515 1.405418 1.8090311.0 0.5 1.0 1.504600 1.608401 1.9474031.0 1.0 0.0 1.433343 1.477003 1.6563531.0 1.0 2.0 1.931369 2.034336 2.398849


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Acknowledgments

The authors are thankful to the reviewers for their constructive suggestions toimprove the quality of the article. One of the authors (K. Bhattacharyya) gratefullyacknowledges the financial support of National Board for Higher Mathematics(NBHM), DAE, Mumbai, India for pursuing this work.

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REACTIVE SOLUTE TRANSFER IN MAGNETOHYDRODYNAMIC BOUNDARY LAYER STAGNATION-POINT FLOW OVER A...

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